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Motion and structure of atmospheric mesoscale baroclinic vortices: dry air and weak environmental shear

Published online by Cambridge University Press:  14 May 2012

Eileen Päschke ,
Patrik Marschalik ,
Antony Z. Owinoh  and
Rupert Klein
Eileen Päschke
Affiliation:
Institut für Mathematik, Technische Universität Berlin, 14195 Berlin, Germany
Patrik Marschalik
Affiliation:
Fachbereich Mathematik & Informatik, Freie Universität Berlin, 10632 Berlin, Germany
Rupert Klein*
Affiliation:
Fachbereich Mathematik & Informatik, Freie Universität Berlin, 10632 Berlin, Germany
*
†Email address for correspondence: [email protected]
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Abstract

A strongly tilted, nearly axisymmetric vortex in dry air with asymmetric diabatic heating is analysed here by matched asymptotic expansions. The vortex is in gradient wind balance, with vortex Rossby numbers of order unity, and embedded in a quasi-geostrophic (QG) background wind with weak vertical shear. With wind speeds of , such vortices correspond to tropical storms or nascent hurricanes according to the Saffir–Simpson scale. For asymmetric heating, nonlinear coupling of the evolution equations for the vortex tilt, its core structure, and its influence on the QG background is found. The theory compares well with the established linear theory of precessing quasi-modes of atmospheric vortices, and it corroborates the relationship between vortex tilt and asymmetric potential temperature and vertical velocity patterns as found by Jones (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 821–851) and Frank & Ritchie (Mon. Weath. Rev., vol. 127, 1999, pp. 2044–2061) in simulations of adiabatic tropical cyclones. A relation between the present theory and the local induction approximation for three-dimensional slender vortex filaments is established.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Meteorology Geophysical and Geological Flows: Quasi-geostrophic flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 701 , 25 June 2012 , pp. 137 - 170
Copyright
Copyright © Cambridge University Press 2012

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Footnotes

‡

Dr Owinoh passed away while this paper was under revision.

References

1. Afanasyev, Ya. D. & Peltier, W. R. 1998 Three-dimensional instability of anticyclonic swirling flow in rotating fluid: laboratory experiments and related theoretical predictions. Phys. Fluids 10 (12), 3194–3202.CrossRefGoogle Scholar
2. Bronstein, I. & Semendjajew, K. 1979 Taschenbuch der Mathematik. Harri Deutsch.Google Scholar
3. Callegari, A. J. & Ting, L. 1978 Motion of a curved vortex filament with decaying vortical core and axial velocity. SIAM J. Appl. Maths 35 (1), 148–175.Google Scholar
4. Charney, J. G. & Eliassen, A. 1964 On the growth of the hurricane depression. J. Atmos. Sci. 21, 68–75.2.0.CO;2>CrossRefGoogle Scholar
5. Eckhaus, W. 1979 Asymptotic Analysis of Singular Perturbations, vol. 9. North-Holland.Google Scholar
6. Egger, J. 1992 Point vortices in a low-order model of barotropic flow on the sphere. Q. J. R. Meteorol. Soc. 118, 533–552.Google Scholar
7. Eliassen, A. 1952 Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norv. 5, 19–60.Google Scholar
8. Emanuel, K. A. 1991 The theory of hurricanes. Annu. Rev. Fluid Mech. 23, 179–196.Google Scholar
9. Emanuel, K. A. 2003 Tropical cyclones. Annu. Rev. Earth Planet. Sci. 31, 75–104.CrossRefGoogle Scholar
10. Frank, W. M. & Ritchie, E. A. 1999 Effects of environmental flow upon tropical cyclone structure. Mon. Weath. Rev. 127, 2044–2061.2.0.CO;2>CrossRefGoogle Scholar
11. Jones, S. 1995 The evolution of vortices in vertical shear i: initially barotropic vortices. Q. J. R. Meteorol. Soc. 121, 821–851.CrossRefGoogle Scholar
12. Jones, S. 2000 The evolution of vortices in vertical shear ii: large-scale asymmetries. Q. J. R. Meteorol. Soc. 126, 3137–3159.Google Scholar
13. Jones, S. 2004 On the ability of dry tropical-cyclone-like vortices to withstand vertical shear. J. Atmos. Sci 61, 114–119.Google Scholar
14. Keller, J. B. & Ward, W. 1996 Asymptotics beyond all orders for a low Reynolds number flow. J. Engng Maths 30, 253–265.CrossRefGoogle Scholar
15. Klein, R. 2010 Scale-dependent asymptotic models for atmospheric flows. Annu. Rev. Fluid Mech. 42, 249–274.CrossRefGoogle Scholar
16. Klein, R. & Majda, A. J. 1991 Self-stretching of a perturbed vortex filament i: the asymptotic equation for deviations from a straight line. Physica D 49, 323–352.CrossRefGoogle Scholar
17. Ling, G. & Ting, L. 1988 Two-time scales inner solutions and motion of a geostrophic vortex. Sci. Sin. XXXI (7).Google Scholar
18. McWilliams, J. C., Graves, L. P. & Montgomery, M. T. 2003 A formal theory for vortex Rossby waves and vortex evolution. Geophys. Astrophys. Fluid Dyn. 97 (4), 275–309.CrossRefGoogle Scholar
19. Mikusky, E. 2007 On the structure of concentrated atmospheric vortices in a gradient wind regime and its motion on synoptic scales. PhD thesis, Universität Hamburg, Fachbereich Geowissenschaften.Google Scholar
20. Montgomery, M. T. & Kallenbach, R. K. 1997 A theory for vortex Rossby waves and its application to spiral bands and intensity changes in hurricanes. Q. J. R. Meteorol. Soc. 123, 435–465.Google Scholar
21. Morikawa, G. K. 1960 Geostrophic vortex motion. J. Meteorol. 17, 148–158.2.0.CO;2>CrossRefGoogle Scholar
22. Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.Google Scholar
23. Plougonven, R. & Zeitlin, V. 2002 Internal gravity wave emission from a pancake vortex: An example of wave–vortex interaction in strongly stratified flows. Phys. Fluids 14, 1259–1268.CrossRefGoogle Scholar
24. Reasor, P. D. & Montgomery, M. T. 2001 Three-dimensional alignment and corotation of weak, tc-like vortices via linear vortex rossby waves. J. Atmos. Sci 58, 2306–2330.2.0.CO;2>CrossRefGoogle Scholar
25. Reasor, P. D., Montgomery, M. T. & Grasso, L. D. 2004 A new look an the problem of tropical cyclones in vertical shear flow. J. Atmos. Sci. 61 (1), 3–22.2.0.CO;2>CrossRefGoogle Scholar
26. Reznik, G. M. 1992 Dynamics of singular vortices on a beta-plane. J. Fluid Mech. 240, 405–432.Google Scholar
27. Reznik, G. M. & Grimshaw, R. 2001 Ageostrophic dynamics of an intense localized vortex on a -plane. J. Fluid Mech. 443, 351–376.Google Scholar
28. Reznik, G. & Kizner, Z. 2007a Two-layer quasi-geostrophic singular vortices embedded in a regular flow. Part 1. Invariants of motion and stability of vortex pairs. J. Fluid Mech. 584, 185–202.CrossRefGoogle Scholar
29. Reznik, G. & Kizner, Z. 2007b Two-layer quasi-geostrophic singular vortices embedded in a regular flow. Part 2. Steady and unsteady drift of individual vortices on a beta-plane. J. Fluid Mech. 584, 203–223.CrossRefGoogle Scholar
30. Ricca, R. L. 1991 Rediscovery of da rios equations. Nature 352, 561–562.Google Scholar
31. Schecter, D. A. & Montgomery, M. T. 2003 On the symmetrization rate of an intense geophysical vortex. Dyn. Atmos. Oceans 37, 55–88.Google Scholar
32. Schecter, D. A. & Montgomery, M. T. 2004 Damping and pumping of a vortex Rossby wave in a monotonic cyclone: critical layer stirring versus inertia–buoyancy wave emission. Phys. Fluids 16, 1334–1348.CrossRefGoogle Scholar
33. Schecter, D. A. & Montgomery, M. T. 2006 Conditions that inhibit the spontaneous radiation of spiral inertia–gravity waves from an intense mesoscale cyclone. J. Atmos. Sci 63, 435–456.CrossRefGoogle Scholar
34. Schecter, D. A. & Montgomery, M. T. 2007 Waves in a cloudy vortex. J. Atmos. Sci 64, 314–337.Google Scholar
35. Schecter, D. A., Montgomery, M. T. & Reasor, P. D. 2002 A theory for the vertical alignment of a quasigeostrophic vortex. J. Atmos. Sci 59, 150–168.Google Scholar
36. Schubert, W. H. & Hack, J. J. 1983 Transformed Eliassen balanced vortex model. J. Atmos. Sci 39, 1687–1697.Google Scholar
37. Shapiro, L. J. & Montgomery, M. T. 1993 A three-dimensional theory for rapidly rotating vortices. J. Atmos. Sci 50, 3322–3335.Google Scholar
38. Smith, R. K. 1991 An analytic theory of tropical-cyclone motion in a barotropic shear flow. Q. J. R. Meteorol. Soc. 47, 685–714.Google Scholar
39. Smith, R. & Montgomery, M. T. 2010 Hurricane boundary-layer theory. Q. J. R. Meteorol. Soc. 136, 1665–1670.Google Scholar
40. Smith, R. K. & Ulrich, W. 1990 An analytic theory of tropical cyclone motion using a barotropic model. J. Atmos. Sci 47, 1973–1986.Google Scholar
41. Stewart, H. J. 1943 Periodic properties of the semi-permanent atmospheric pressure systems. Q. Appl. Maths 1, 262–267.CrossRefGoogle Scholar
42. Ting, L., Klein, R. & Knio, O. M. 2007 Vortex Dominated Flows: Analysis and Computation for Multiple Scales, Series in Applied Mathematical Sciences , vol. 116. Springer.Google Scholar
43. Ting, L. & Ling, G. 1983 Studies on the motion and core structure of a geostrophic vortex. In Proc. 2nd Asian Congress of Fluid Mechanics, pp. 900–905. Science Press.Google Scholar
44. Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar
45. Wirth, V. & Dunkerton, T. J. 2009 The dynamics of eye formation and maintenance in axisymmetric diabatic vortices. J. Atmos. Sci 66, 3601–3620.CrossRefGoogle Scholar